Answer to September 23, 2002 Problem

Road Trip To Pennsylvania

 
A certain Math Teacher traveled from Winchester, Virginia to Harrisburg, Pennsylvania at a constant rate of 25 mph. How fast must she travel on the return trip so that she averages 50 mph over the entire trip?

(The distance from Harrisburg to Winchester is not relevant to the problem. You only need to know that the distance from Winchester to Harrisburg is the same as the distance from Harrisburg to Winchester.)

 

Solution to the Problem:

It is not possible to go fast enough so that the teacher will average 50 mph over the entire trip.

First, it is necessary to realize that "average speed" is defined to be total distance traveled divided by the total time it takes to get there.

To see that it is not possible to go fast enough to average 50 mph, let the distance from Winchester to Harrisburg be 100 miles (any number will do, or you can use a variable for this).
Traveling at 25 mph, it would take 4 hours to get to Harrisburg.
In order to average 50 mph over the entire trip of 200 miles, you would need to make the trip in 200 / 50 = 4 hours.
But that can't be done unless you make the return trip in 0 seconds!!!

  Rate Time Distance
Winchester to Harrisburg 25 mph 4 hours 100 miles
Harrisburg to Winchester --- mph 0 hours 100 miles
Entire Trip 50 mph 4 hours 200 miles

The most common wrong answer was 75 mph.
But if you returned at 75 mph, you would average only 37.5 mph over the entire trip (NOT 50 mph)! Remember that average speed equals the total distance traveled divided by the total time. Look at the Rate-Time-Distance table below to see why 75 mph is incorrect:

  Rate Time Distance
Winchester to Harrisburg 25 mph 4 hours 100 miles
Harrisburg to Winchester 75 mph 1 1/3 hours 100 miles
Entire Trip 37.5 mph 5 1/3 hours 200 miles


Correctly solved by:

1. Walt Arrison Philadelphia, Pennsylvania
2. Rick Jones Kennett Square, Pennsylvania
3. Richard K. Johnson La Jolla, California
4. David and Judy Dixon Bennettsville, South Carolina
5. Kirstine Wynn St. Olaf's College
Northfield, Minnesota
6. James Alarie University of Michigan -- Flint
Flint, Michigan
7. Izzy Kushner Closter, New Jersey
8. Tina Zahel Winchester, Virginia
9. Frank Maire Norwalk, Ohio
10. Tom Hanzl Brno, Czech republic
11. Jeff Gaither Winchester, Virginia
12. Geoff Keith Santa Monica, California
13. Kevin Thompson Oregon
14. Daniel Wilberger Winchester, Virginia
15. Peggah Sadeghzadeh Winchester, Virginia
16. Torbjorn Jansson Garphyttan, Sweden
17. Steven Strasburg Memphis, Tennessee
18. George Gaither Winchester, Virginia
19. Kathleen Altemose Winchester, Virginia
20. Matt Stewart Columbus, Georgia
21. Richard Mocarski University of Michigan
Ann Arbor, Michigan
22. Paul E. Haak ----------